A study investigated the impact of smoking on sleep patterns by measuring the time taken to fall asleep in minutes. The following data were gathered: Smokers: 69.3, 56.0, 22.1, 47.6, 53.2, 48.1, 52.7, 34.4, 60.2, 43.8, 23.2, 13.8 Nonsmokers: 28.6, 25.1, 26.4, 34.9, 29.8, 28.4, 38.5, 30.2, 30.6, 31.8, 41.6, 21.1, 36.0, 37.9, 13.9 (a) Calculate the sample mean for each group. (b) Calculate the sample standard deviation for each group. (c) Construct a dot plot for both data sets on the same axis. (d) Discuss the potential effect of smoking on the time it takes to fall asleep.

(a) To calculate the sample mean, we add up all the values for each group and divide by the number of observations in each group.

Smokers' mean:

(69.3 + 56.0 + 22.1 + 47.6 + 53.2 + 48.1 + 52.7 + 34.4 + 60.2 + 43.8 + 23.2 + 13.8) / 12 = 524.4 / 12 ≈ 43.7 minutes

Nonsmokers' mean:

(28.6 + 25.1 + 26.4 + 34.9 + 29.8 + 28.4 + 38.5 + 30.2 + 30.6 + 31.8 + 41.6 + 21.1 + 36.0 + 37.9 + 13.9) / 15 = 464.9 / 15 ≈ 31.0 minutes

(b) To calculate the sample standard deviation, first we calculate the variance. Variance is the average squared deviation of each number from the mean. Then, the standard deviation is the square root of the variance.

For smokers: Mean = 43.7 Variance = [(69.3 - 43.7)^2 + (56.0 - 43.7)^2 + ... + (13.8 - 43.7)^2] / (12 - 1) Variance ≈ 341.686 Standard deviation for smokers ≈ √341.686 ≈ 18.5 minutes

For nonsmokers: Mean = 31.0 Variance = [(28.6 - 31.0)^2 + (25.1 - 31.0)^2 + ... + (13.9 - 31.0)^2] / (15 - 1) Variance ≈ 57.907 Standard deviation for nonsmokers ≈ √57.907 ≈ 7.6 minutes

(c) Constructing a dot plot requires a bit of graphical work that cannot be represented in text. However, I can explain how you would go about creating one:

1. Draw a horizontal line to represent the axis and mark it with time taken to fall asleep in minutes. 2. Above this axis at the appropriate time positions, place dots for each smoker's time in one colour. 3. Below the axis at the appropriate time positions, do the same for nonsmokers' times in another colour. 4. Each dot represents one observation from the respective dataset. 5. The dots for similar values are stacked vertically.

(d) Looking at the mean times to fall asleep for each group, we can see that smokers have a higher average time (43.7 minutes) than nonsmokers (31.0 minutes). This suggests that smokers may take longer to fall asleep on average. When comparing the standard deviations, smokers also vary more in the time it takes them to fall asleep, as indicated by a higher standard deviation (18.5 minutes for smokers versus 7.6 minutes for nonsmokers). This increased variation could be a sign of the disruptive effects of smoking on sleep patterns.

It is key to note that whilst this data may indicate a correlation between smoking and increased time to fall asleep, it does not necessarily imply causation. Other factors could be influencing sleep times, and further research would be needed to make a definitive conclusion.

Extra: In statistics, mean and standard deviation are measures of central tendency and dispersion, respectively. They provide a summary of the data. The mean gives the average of the data points, while the standard deviation measures the amount of variation or dispersion from the average.

The dot plot is a simple graphical representation that provides a visual impression of the data distribution, showing how frequently data points appear and any clustering around certain values.

When assessing the potential impact of a variable like smoking on sleep patterns, researchers use statistical analyses to identify patterns in the data. Mean and standard deviation are often the first steps to summarize data; however, understanding causality would require additional analysis, such as controlling for confounding variables or establishing temporality - all part of more sophisticated research designs such as randomized controlled trials or longitudinal studies.